Properties

Label 1710.2.a.h.1.1
Level $1710$
Weight $2$
Character 1710.1
Self dual yes
Analytic conductor $13.654$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1710.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.6544187456\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1710.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -2.00000 q^{11} -4.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{19} +1.00000 q^{20} +2.00000 q^{22} +8.00000 q^{23} +1.00000 q^{25} +4.00000 q^{26} -2.00000 q^{28} -6.00000 q^{29} -8.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} -2.00000 q^{35} -8.00000 q^{37} -1.00000 q^{38} -1.00000 q^{40} -12.0000 q^{41} -2.00000 q^{44} -8.00000 q^{46} -3.00000 q^{49} -1.00000 q^{50} -4.00000 q^{52} +10.0000 q^{53} -2.00000 q^{55} +2.00000 q^{56} +6.00000 q^{58} -6.00000 q^{59} -6.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +12.0000 q^{67} +6.00000 q^{68} +2.00000 q^{70} +12.0000 q^{71} -10.0000 q^{73} +8.00000 q^{74} +1.00000 q^{76} +4.00000 q^{77} -8.00000 q^{79} +1.00000 q^{80} +12.0000 q^{82} -8.00000 q^{83} +6.00000 q^{85} +2.00000 q^{88} -8.00000 q^{89} +8.00000 q^{91} +8.00000 q^{92} +1.00000 q^{95} -14.0000 q^{97} +3.00000 q^{98} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 12.0000 1.32518
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 3.00000 0.303046
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 18.0000 1.59724 0.798621 0.601834i \(-0.205563\pi\)
0.798621 + 0.601834i \(0.205563\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 4.00000 0.350823
\(131\) 2.00000 0.174741 0.0873704 0.996176i \(-0.472154\pi\)
0.0873704 + 0.996176i \(0.472154\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) 8.00000 0.668994
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 10.0000 0.827606
\(147\) 0 0
\(148\) −8.00000 −0.657596
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) −4.00000 −0.322329
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) 24.0000 1.91541 0.957704 0.287754i \(-0.0929087\pi\)
0.957704 + 0.287754i \(0.0929087\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −12.0000 −0.937043
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) −6.00000 −0.460179
\(171\) 0 0
\(172\) 0 0
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) 8.00000 0.599625
\(179\) 2.00000 0.149487 0.0747435 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) −8.00000 −0.592999
\(183\) 0 0
\(184\) −8.00000 −0.589768
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) −12.0000 −0.877527
\(188\) 0 0
\(189\) 0 0
\(190\) −1.00000 −0.0725476
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −26.0000 −1.85242 −0.926212 0.377004i \(-0.876954\pi\)
−0.926212 + 0.377004i \(0.876954\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −2.00000 −0.140720
\(203\) 12.0000 0.842235
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 14.0000 0.975426
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 10.0000 0.686803
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) −2.00000 −0.134840
\(221\) −24.0000 −1.61441
\(222\) 0 0
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) 16.0000 1.06196 0.530979 0.847385i \(-0.321824\pi\)
0.530979 + 0.847385i \(0.321824\pi\)
\(228\) 0 0
\(229\) 30.0000 1.98246 0.991228 0.132164i \(-0.0421925\pi\)
0.991228 + 0.132164i \(0.0421925\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) 0 0
\(238\) 12.0000 0.777844
\(239\) −28.0000 −1.81117 −0.905585 0.424165i \(-0.860568\pi\)
−0.905585 + 0.424165i \(0.860568\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 7.00000 0.449977
\(243\) 0 0
\(244\) −6.00000 −0.384111
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) −16.0000 −1.00591
\(254\) −18.0000 −1.12942
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −10.0000 −0.623783 −0.311891 0.950118i \(-0.600963\pi\)
−0.311891 + 0.950118i \(0.600963\pi\)
\(258\) 0 0
\(259\) 16.0000 0.994192
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) −2.00000 −0.123560
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 10.0000 0.614295
\(266\) 2.00000 0.122628
\(267\) 0 0
\(268\) 12.0000 0.733017
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) −2.00000 −0.120605
\(276\) 0 0
\(277\) 4.00000 0.240337 0.120168 0.992754i \(-0.461657\pi\)
0.120168 + 0.992754i \(0.461657\pi\)
\(278\) 20.0000 1.19952
\(279\) 0 0
\(280\) 2.00000 0.119523
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) 24.0000 1.41668
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) −10.0000 −0.585206
\(293\) 30.0000 1.75262 0.876309 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) 8.00000 0.464991
\(297\) 0 0
\(298\) 2.00000 0.115857
\(299\) −32.0000 −1.85061
\(300\) 0 0
\(301\) 0 0
\(302\) −20.0000 −1.15087
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) −6.00000 −0.343559
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 4.00000 0.227921
\(309\) 0 0
\(310\) 8.00000 0.454369
\(311\) 20.0000 1.13410 0.567048 0.823685i \(-0.308085\pi\)
0.567048 + 0.823685i \(0.308085\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −24.0000 −1.35440
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 16.0000 0.891645
\(323\) 6.00000 0.333849
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) 12.0000 0.662589
\(329\) 0 0
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −8.00000 −0.439057
\(333\) 0 0
\(334\) 16.0000 0.875481
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) −3.00000 −0.163178
\(339\) 0 0
\(340\) 6.00000 0.325396
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) −2.00000 −0.107521
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) 2.00000 0.106600
\(353\) −10.0000 −0.532246 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) −8.00000 −0.423999
\(357\) 0 0
\(358\) −2.00000 −0.105703
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −14.0000 −0.735824
\(363\) 0 0
\(364\) 8.00000 0.419314
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) 14.0000 0.730794 0.365397 0.930852i \(-0.380933\pi\)
0.365397 + 0.930852i \(0.380933\pi\)
\(368\) 8.00000 0.417029
\(369\) 0 0
\(370\) 8.00000 0.415900
\(371\) −20.0000 −1.03835
\(372\) 0 0
\(373\) 28.0000 1.44979 0.724893 0.688862i \(-0.241889\pi\)
0.724893 + 0.688862i \(0.241889\pi\)
\(374\) 12.0000 0.620505
\(375\) 0 0
\(376\) 0 0
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 1.00000 0.0512989
\(381\) 0 0
\(382\) −12.0000 −0.613973
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 2.00000 0.101797
\(387\) 0 0
\(388\) −14.0000 −0.710742
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) 48.0000 2.42746
\(392\) 3.00000 0.151523
\(393\) 0 0
\(394\) 26.0000 1.30986
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) 20.0000 1.00377 0.501886 0.864934i \(-0.332640\pi\)
0.501886 + 0.864934i \(0.332640\pi\)
\(398\) 24.0000 1.20301
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) 32.0000 1.59403
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) −12.0000 −0.595550
\(407\) 16.0000 0.793091
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 12.0000 0.592638
\(411\) 0 0
\(412\) −14.0000 −0.689730
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) −8.00000 −0.392705
\(416\) 4.00000 0.196116
\(417\) 0 0
\(418\) 2.00000 0.0978232
\(419\) 10.0000 0.488532 0.244266 0.969708i \(-0.421453\pi\)
0.244266 + 0.969708i \(0.421453\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) −10.0000 −0.485643
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 12.0000 0.580721
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 8.00000 0.382692
\(438\) 0 0
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 2.00000 0.0953463
\(441\) 0 0
\(442\) 24.0000 1.14156
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) −8.00000 −0.379236
\(446\) 14.0000 0.662919
\(447\) 0 0
\(448\) −2.00000 −0.0944911
\(449\) −8.00000 −0.377543 −0.188772 0.982021i \(-0.560451\pi\)
−0.188772 + 0.982021i \(0.560451\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) −18.0000 −0.846649
\(453\) 0 0
\(454\) −16.0000 −0.750917
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −30.0000 −1.40181
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 10.0000 0.464739 0.232370 0.972628i \(-0.425352\pi\)
0.232370 + 0.972628i \(0.425352\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) 32.0000 1.48078 0.740392 0.672176i \(-0.234640\pi\)
0.740392 + 0.672176i \(0.234640\pi\)
\(468\) 0 0
\(469\) −24.0000 −1.10822
\(470\) 0 0
\(471\) 0 0
\(472\) 6.00000 0.276172
\(473\) 0 0
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) −12.0000 −0.550019
\(477\) 0 0
\(478\) 28.0000 1.28069
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) 32.0000 1.45907
\(482\) 18.0000 0.819878
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −14.0000 −0.635707
\(486\) 0 0
\(487\) −26.0000 −1.17817 −0.589086 0.808070i \(-0.700512\pi\)
−0.589086 + 0.808070i \(0.700512\pi\)
\(488\) 6.00000 0.271607
\(489\) 0 0
\(490\) 3.00000 0.135526
\(491\) 18.0000 0.812329 0.406164 0.913800i \(-0.366866\pi\)
0.406164 + 0.913800i \(0.366866\pi\)
\(492\) 0 0
\(493\) −36.0000 −1.62136
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −24.0000 −1.07655
\(498\) 0 0
\(499\) −12.0000 −0.537194 −0.268597 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −18.0000 −0.803379
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 16.0000 0.711287
\(507\) 0 0
\(508\) 18.0000 0.798621
\(509\) −14.0000 −0.620539 −0.310270 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) 0 0
\(511\) 20.0000 0.884748
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 10.0000 0.441081
\(515\) −14.0000 −0.616914
\(516\) 0 0
\(517\) 0 0
\(518\) −16.0000 −0.703000
\(519\) 0 0
\(520\) 4.00000 0.175412
\(521\) −24.0000 −1.05146 −0.525730 0.850652i \(-0.676208\pi\)
−0.525730 + 0.850652i \(0.676208\pi\)
\(522\) 0 0
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) 2.00000 0.0873704
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) −48.0000 −2.09091
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) −10.0000 −0.434372
\(531\) 0 0
\(532\) −2.00000 −0.0867110
\(533\) 48.0000 2.07911
\(534\) 0 0
\(535\) 4.00000 0.172935
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) −10.0000 −0.431131
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −6.00000 −0.257248
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −6.00000 −0.256307
\(549\) 0 0
\(550\) 2.00000 0.0852803
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) −4.00000 −0.169944
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) 0 0
\(563\) 16.0000 0.674320 0.337160 0.941447i \(-0.390534\pi\)
0.337160 + 0.941447i \(0.390534\pi\)
\(564\) 0 0
\(565\) −18.0000 −0.757266
\(566\) 20.0000 0.840663
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 8.00000 0.334497
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) 8.00000 0.333623
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −19.0000 −0.790296
\(579\) 0 0
\(580\) −6.00000 −0.249136
\(581\) 16.0000 0.663792
\(582\) 0 0
\(583\) −20.0000 −0.828315
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) −30.0000 −1.23929
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 6.00000 0.247016
\(591\) 0 0
\(592\) −8.00000 −0.328798
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) −2.00000 −0.0819232
\(597\) 0 0
\(598\) 32.0000 1.30858
\(599\) 8.00000 0.326871 0.163436 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 0 0
\(601\) 30.0000 1.22373 0.611863 0.790964i \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 20.0000 0.813788
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 6.00000 0.242933
\(611\) 0 0
\(612\) 0 0
\(613\) −20.0000 −0.807792 −0.403896 0.914805i \(-0.632344\pi\)
−0.403896 + 0.914805i \(0.632344\pi\)
\(614\) −16.0000 −0.645707
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) −12.0000 −0.482321 −0.241160 0.970485i \(-0.577528\pi\)
−0.241160 + 0.970485i \(0.577528\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) −20.0000 −0.801927
\(623\) 16.0000 0.641026
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −14.0000 −0.559553
\(627\) 0 0
\(628\) 24.0000 0.957704
\(629\) −48.0000 −1.91389
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 8.00000 0.318223
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) 18.0000 0.714308
\(636\) 0 0
\(637\) 12.0000 0.475457
\(638\) −12.0000 −0.475085
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 28.0000 1.10593 0.552967 0.833203i \(-0.313496\pi\)
0.552967 + 0.833203i \(0.313496\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) −2.00000 −0.0782660 −0.0391330 0.999234i \(-0.512460\pi\)
−0.0391330 + 0.999234i \(0.512460\pi\)
\(654\) 0 0
\(655\) 2.00000 0.0781465
\(656\) −12.0000 −0.468521
\(657\) 0 0
\(658\) 0 0
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 0 0
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) 28.0000 1.08825
\(663\) 0 0
\(664\) 8.00000 0.310460
\(665\) −2.00000 −0.0775567
\(666\) 0 0
\(667\) −48.0000 −1.85857
\(668\) −16.0000 −0.619059
\(669\) 0 0
\(670\) −12.0000 −0.463600
\(671\) 12.0000 0.463255
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) −6.00000 −0.231111
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 28.0000 1.07454
\(680\) −6.00000 −0.230089
\(681\) 0 0
\(682\) −16.0000 −0.612672
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) −6.00000 −0.229248
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) 0 0
\(689\) −40.0000 −1.52388
\(690\) 0 0
\(691\) −36.0000 −1.36950 −0.684752 0.728776i \(-0.740090\pi\)
−0.684752 + 0.728776i \(0.740090\pi\)
\(692\) 2.00000 0.0760286
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) −20.0000 −0.758643
\(696\) 0 0
\(697\) −72.0000 −2.72719
\(698\) 22.0000 0.832712
\(699\) 0 0
\(700\) −2.00000 −0.0755929
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 10.0000 0.376355
\(707\) −4.00000 −0.150435
\(708\) 0 0
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) −12.0000 −0.450352
\(711\) 0 0
\(712\) 8.00000 0.299813
\(713\) −64.0000 −2.39682
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) 2.00000 0.0747435
\(717\) 0 0
\(718\) −8.00000 −0.298557
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) 14.0000 0.520306
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) −22.0000 −0.815935 −0.407967 0.912996i \(-0.633762\pi\)
−0.407967 + 0.912996i \(0.633762\pi\)
\(728\) −8.00000 −0.296500
\(729\) 0 0
\(730\) 10.0000 0.370117
\(731\) 0 0
\(732\) 0 0
\(733\) 36.0000 1.32969 0.664845 0.746981i \(-0.268498\pi\)
0.664845 + 0.746981i \(0.268498\pi\)
\(734\) −14.0000 −0.516749
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) −24.0000 −0.884051
\(738\) 0 0
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) −8.00000 −0.294086
\(741\) 0 0
\(742\) 20.0000 0.734223
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) −2.00000 −0.0732743
\(746\) −28.0000 −1.02515
\(747\) 0 0
\(748\) −12.0000 −0.438763
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −24.0000 −0.874028
\(755\) 20.0000 0.727875
\(756\) 0 0
\(757\) −36.0000 −1.30844 −0.654221 0.756303i \(-0.727003\pi\)
−0.654221 + 0.756303i \(0.727003\pi\)
\(758\) −12.0000 −0.435860
\(759\) 0 0
\(760\) −1.00000 −0.0362738
\(761\) 28.0000 1.01500 0.507500 0.861652i \(-0.330570\pi\)
0.507500 + 0.861652i \(0.330570\pi\)
\(762\) 0 0
\(763\) 4.00000 0.144810
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) −4.00000 −0.144150
\(771\) 0 0
\(772\) −2.00000 −0.0719816
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) 26.0000 0.932145
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) −24.0000 −0.858788
\(782\) −48.0000 −1.71648
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 24.0000 0.856597
\(786\) 0 0
\(787\) −12.0000 −0.427754 −0.213877 0.976861i \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) −26.0000 −0.926212
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) 36.0000 1.28001
\(792\) 0 0
\(793\) 24.0000 0.852265
\(794\) −20.0000 −0.709773
\(795\) 0 0
\(796\) −24.0000 −0.850657
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −24.0000 −0.847469
\(803\) 20.0000 0.705785
\(804\) 0 0
\(805\) −16.0000 −0.563926
\(806\) −32.0000 −1.12715
\(807\) 0 0
\(808\) −2.00000 −0.0703598
\(809\) −16.0000 −0.562530 −0.281265 0.959630i \(-0.590754\pi\)
−0.281265 + 0.959630i \(0.590754\pi\)
\(810\) 0 0
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 12.0000 0.421117
\(813\) 0 0
\(814\) −16.0000 −0.560800
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) 0 0
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) −38.0000 −1.32621 −0.663105 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) 0 0
\(823\) 26.0000 0.906303 0.453152 0.891434i \(-0.350300\pi\)
0.453152 + 0.891434i \(0.350300\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) 48.0000 1.66912 0.834562 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) 0 0
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 8.00000 0.277684
\(831\) 0 0
\(832\) −4.00000 −0.138675
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) −2.00000 −0.0691714
\(837\) 0 0
\(838\) −10.0000 −0.345444
\(839\) 16.0000 0.552381 0.276191 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −10.0000 −0.344623
\(843\) 0 0
\(844\) 4.00000 0.137686
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) 10.0000 0.343401
\(849\) 0 0
\(850\) −6.00000 −0.205798
\(851\) −64.0000 −2.19389
\(852\) 0 0
\(853\) −12.0000 −0.410872 −0.205436 0.978671i \(-0.565861\pi\)
−0.205436 + 0.978671i \(0.565861\pi\)
\(854\) −12.0000 −0.410632
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) −38.0000 −1.29806 −0.649028 0.760765i \(-0.724824\pi\)
−0.649028 + 0.760765i \(0.724824\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 40.0000 1.36162 0.680808 0.732462i \(-0.261629\pi\)
0.680808 + 0.732462i \(0.261629\pi\)
\(864\) 0 0
\(865\) 2.00000 0.0680020
\(866\) 34.0000 1.15537
\(867\) 0 0
\(868\) 16.0000 0.543075
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) −48.0000 −1.62642
\(872\) 2.00000 0.0677285
\(873\) 0 0
\(874\) −8.00000 −0.270604
\(875\) −2.00000 −0.0676123
\(876\) 0 0
\(877\) 12.0000 0.405211 0.202606 0.979260i \(-0.435059\pi\)
0.202606 + 0.979260i \(0.435059\pi\)
\(878\) 28.0000 0.944954
\(879\) 0 0
\(880\) −2.00000 −0.0674200
\(881\) −52.0000 −1.75192 −0.875962 0.482380i \(-0.839773\pi\)
−0.875962 + 0.482380i \(0.839773\pi\)
\(882\) 0 0
\(883\) −36.0000 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) −24.0000 −0.807207
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 0 0
\(889\) −36.0000 −1.20740
\(890\) 8.00000 0.268161
\(891\) 0 0
\(892\) −14.0000 −0.468755
\(893\) 0 0
\(894\) 0 0
\(895\) 2.00000 0.0668526
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) 8.00000 0.266963
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) 60.0000 1.99889
\(902\) −24.0000 −0.799113
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) 8.00000 0.265636 0.132818 0.991140i \(-0.457597\pi\)
0.132818 + 0.991140i \(0.457597\pi\)
\(908\) 16.0000 0.530979
\(909\) 0 0
\(910\) −8.00000 −0.265197
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 30.0000 0.991228
\(917\) −4.00000 −0.132092
\(918\) 0 0
\(919\) −20.0000 −0.659739 −0.329870 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) −8.00000 −0.263752
\(921\) 0 0
\(922\) −30.0000 −0.987997
\(923\) −48.0000 −1.57994
\(924\) 0 0
\(925\) −8.00000 −0.263038
\(926\) −10.0000 −0.328620
\(927\) 0 0
\(928\) 6.00000 0.196960
\(929\) 36.0000 1.18112 0.590561 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) −10.0000 −0.327561
\(933\) 0 0
\(934\) −32.0000 −1.04707
\(935\) −12.0000 −0.392442
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 24.0000 0.783628
\(939\) 0 0
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 0 0
\(943\) −96.0000 −3.12619
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) 40.0000 1.29845
\(950\) −1.00000 −0.0324443
\(951\) 0 0
\(952\) 12.0000 0.388922
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) 12.0000 0.388311
\(956\) −28.0000 −0.905585
\(957\) 0 0
\(958\) 12.0000 0.387702
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −32.0000 −1.03172
\(963\) 0 0
\(964\) −18.0000 −0.579741
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) −2.00000 −0.0643157 −0.0321578 0.999483i \(-0.510238\pi\)
−0.0321578 + 0.999483i \(0.510238\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) 14.0000 0.449513
\(971\) −10.0000 −0.320915 −0.160458 0.987043i \(-0.551297\pi\)
−0.160458 + 0.987043i \(0.551297\pi\)
\(972\) 0 0
\(973\) 40.0000 1.28234
\(974\) 26.0000 0.833094
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) 26.0000 0.831814 0.415907 0.909407i \(-0.363464\pi\)
0.415907 + 0.909407i \(0.363464\pi\)
\(978\) 0 0
\(979\) 16.0000 0.511362
\(980\) −3.00000 −0.0958315
\(981\) 0 0
\(982\) −18.0000 −0.574403
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) −26.0000 −0.828429
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) −4.00000 −0.127257
\(989\) 0 0
\(990\) 0 0
\(991\) −60.0000 −1.90596 −0.952981 0.303029i \(-0.902002\pi\)
−0.952981 + 0.303029i \(0.902002\pi\)
\(992\) 8.00000 0.254000
\(993\) 0 0
\(994\) 24.0000 0.761234
\(995\) −24.0000 −0.760851
\(996\) 0 0
\(997\) −56.0000 −1.77354 −0.886769 0.462213i \(-0.847056\pi\)
−0.886769 + 0.462213i \(0.847056\pi\)
\(998\) 12.0000 0.379853
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1710.2.a.h.1.1 1
3.2 odd 2 1710.2.a.m.1.1 yes 1
5.4 even 2 8550.2.a.bg.1.1 1
15.14 odd 2 8550.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1710.2.a.h.1.1 1 1.1 even 1 trivial
1710.2.a.m.1.1 yes 1 3.2 odd 2
8550.2.a.p.1.1 1 15.14 odd 2
8550.2.a.bg.1.1 1 5.4 even 2